Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.